Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2011, Article ID 790942, 9 pages
doi:10.1155/2011/790942
Research Article
Normal Criteria of Function Families Concerning
Shared Values
Wenjun Yuan,1 Bing Zhu,2 and Jianming Lin3
1
School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China
College of Computer Engineering Technology, Guangdong Institute of Science and Technology,
Zhuhai 519090, China
3
School of Economic and Management, Guangzhou University of Chinese Medicine,
Guangzhou 510006, China
2
Correspondence should be addressed to Jianming Lin, ljmguanli@21cn.com
Received 10 July 2011; Accepted 7 September 2011
Academic Editor: Hui-Shen Shen
Copyright q 2011 Wenjun Yuan et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
We study the normality of families of meromorphic functions concerning shared values. We
consider whether a family of meromorphic functions F is normal in D, if, for every pair of functions
f and g in F, f −af −n and g −ag −n share the value b, where a and b are two finite complex numbers
such that a /
0, n is a positive integer. Some examples show that the conditions in our results are
best possible.
1. Introduction and Main Results
Let fz and gz be two nonconstant meromorphic functions in a domain D ⊆ C, and let a
be a finite complex value. We say that f and g share a CM or IM in D provided that f − a
and g − a have the same zeros counting or ignoring multiplicity in D. When a ∞, the
zeros of f − a mean the poles of f see 1. It is assumed that the reader is familiar with the
standard notations and the basic results of Nevanlinna’s value-distribution theory 2–4 or
1.
Bloch’s principle 5 states that every condition which reduces a meromorphic
function in the plane C to be a constant forces a family of meromorphic functions in a domain
D to be normal. Although the principle is false in general see 6, many authors proved
normality criterion for families of meromorphic functions corresponding to Liouville-Picard
type theorem see 7 or 4.
It is also more interesting to find normality criteria from the point of view of
shared values. In this area, Schwick 8 first proved an interesting result that a family of
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Journal of Applied Mathematics
meromorphic functions in a domain is normal if every function shares three distinct finite
complex numbers with its first derivative. And later, more results about normality criteria
concerning shared values can be found, for instance, in 9–11 and so on. In recent years, this
subject has attracted the attention of many researchers worldwide.
We now first introduce a normality criterion related to a Hayman normal conjecture
12.
Theorem 1.1. Let F be a family of holomorphic (meromorphic) functions defined in a domain D,
n ∈ N, a / 0 for each function fz ∈ F and n ≥ 2 n ≥ 3, then F
/ 0, b ∈ C. If f z af n z − b is normal in D.
The results for the holomorphic case are due to Drasin 7 for n ≥ 3, Pang 13 for n 3,
Chen and Fang 14 for n 2, Ye 15 for n 2, and Chen and Gu 16 for the generalized
result with a and b replaced by meromorphic functions. The results for the meromorphic case
are due to Li 17, Li 18 and Langley 19 for n ≥ 5, Pang 13 for n 4, Chen and Fang 14
for n 3, and Zalcman 20 for n 3, obtained independently.
When n 2 and F is meromorphic, Theorem 1.1 is not valid in general. Fang and Yuan
21 gave an example to this, and moreover a result added other conditions below.
Example 1.2. The family of meromorphic functions F {fj z jz/ jz − 12 : j 1, 2, . . . , }
is not normal in D {z : |z| < 1}. This is deduced by fj# 0 j → ∞, as j → ∞ and Marty’s
criterion 2, although, for any fj z ∈ F, fj fj2 j jz − 1−4 / 0.
Here f # ξ denotes the spherical derivative
f ξ
f ξ 2 .
1 fξ
#
1.1
Theorem 1.3. Let F be a family of meromorphic functions in a domain D, and a /
0, b ∈ C. If
f z afz2 − b /
0 and the poles of fz are of multiplicity ≥3 for each fz ∈ F, then F is
normal in D.
In 2008, by the ideas of shared values, Zhang 11 proved the following.
Theorem 1.4. Let F be a family of meromorphic (holomorphic) functions in D, n a positive integer,
and a, b two finite complex numbers such that a /
0. If n ≥ 4 n ≥ 2 and, for every pair of functions
f and g in F, f − af n and g − ag n share the value b, then F is normal in D.
Example 1.5 see 11. The family of meromorphic functions F {fj z 1/ jz − 1/j :
j 1, 2, . . . , } is not normal in D {z : |z| < 1}. Obviously fj − fj3 −z/ jz − 1/j3 . So
3
− fm
share the value 0 in D, but F is not normal at the point
for each pair m, j, fj − fj3 and fm
3
#
z 0, since fj 0 2 j /1 j → ∞, as j → ∞.
Remark 1.6. Example 1.5 shows that Theorem 1.4 is not valid when n 3, and the condition
n 4 is best possible for meromorphic case.
Journal of Applied Mathematics
3
In this paper, we will consider the similar relations and prove the following results.
Theorem 1.7. Let F be a family of meromorphic functions in D, n a positive integer, and a, b two
finite complex numbers such that a /
0. If n ≥ 2 and, for every pair of functions f and g in F, f −af −n
−n
and g − ag share the value b, then F is normal in D.
Example 1.8. The family of holomorphic functions F {fj z jz − 1/j : j 1, 2, . . . , }
is not normal in D {z : |z| < 1}. This is deduced by fj# 0 j j/j 1 → ∞, as j → ∞
and Marty’s criterion 2, although, for any fj z ∈ F, fj fj−1 j jz/jz − 1.
Remark 1.9. Example 1.8 shows that the condition that added n ≥ 2 in Theorem 1.7 is best
possible. In Theorem 1.7 taking b 0 we get Corollary 1.10 obtained by Zhang 22.
Corollary 1.10. Let F be a family of meromorphic functions in D, n ≥ 2, and let a be a nonzero finite
complex number. If, for every pair of functions f and g in F, f n f and g n g share the value a, then F
is normal in D.
A natural problem is what conditions are added such that Theorem 1.7 holds when
n 1. Next we give an answer.
Theorem 1.11. Let F be a family of meromorphic functions in D, and let a and b be two finite complex
numbers such that a /
0. Suppose that all of zeros are multiple for each fz ∈ F. If, for every pair of
functions f and g in F, f − af −1 and g − ag −1 share the value b, then F is normal in D.
Remark 1.12. Example 1.8 shows that the condition that all of zeros are multiple for each
fz ∈ F added in Theorem 1.7 is best possible. In Theorem 1.11 taking b 0 we get
Corollary 1.13.
Corollary 1.13. Let F be a family of meromorphic functions in D, and let a be a nonzero finite complex
number. Suppose that all of zeros are multiple for each fz ∈ F. If, for every pair of functions f and
g in F, ff and gg share the value a, then F is normal in D.
From the proof of Theorem 1.7 we know that the following corollary holds.
Corollary 1.14. Let F be a family of meromorphic functions in D, n be a positive integer and a, b
be two finite complex numbers such that a /
0. If for each function f in F, f − af −n /
b, then F is
normal in D.
2. Preliminary Lemmas
In order to prove our result, we need the following lemmas. The first one extends a famous
result by Zalcman 23 concerning normal families.
Lemma 2.1 see 24. Let F be a family of meromorphic functions on the unit disc satisfying all
zeros of functions in F that have multiplicity ≥ p and all poles of functions in F that have multiplicity
≥ q. Let α be a real number satisfying −q < α < p. Then F is not normal at 0 if and only if there exist
a a number 0 < r < 1;
b points zn with |zn | < r;
4
Journal of Applied Mathematics
c functions fn ∈ F;
d positive numbers ρn → 0
such that gn ζ : ρ−α fn zn ρn ζ converges spherically uniformly on each compact subset of C to
a nonconstant meromorphic function gζ, whose all zeros have multiplicity ≥ p and all poles have
multiplicity ≥ q and order is at most 2.
Remark 2.2. If F is a family of holomorphic functions on the unit disc in Lemma 2.1, then gζ
is a nonconstant entire function whose order is at most 1.
The order of g is defined by using Nevanlinna’s characteristic function T r, g:
log T r, g
.
ρ g lim sup
r →∞
log r
2.1
Lemma 2.3 see 25 or 26. Let fz be a meromorphic function and c ∈ C \ {0}. If fz has
neither simple zero nor simple pole, and f z / c, then fz is constant.
Lemma 2.4 see 27. Let fz be a transcendental meromorphic function of finite order in C and
have no simple zero, then f z assumes every nonzero finite value infinitely often.
3. Proof of the Results
Proof of Theorem 1.7. Suppose that F is not normal in D. Then there exists at least one point z0
such that F is not normal at the point z0 . Without loss of generality we assume that z0 0. By
Lemma 2.1, there exist points zj → 0, positive numbers ρj → 0, and functions fj ∈ F such
that
−1/n1
gj ξ ρj
fj zj ρj ξ ⇒ gξ
3.1
locally uniformly with respect to the spherical metric, where g is a nonconstant meromorphic
function in C. Moreover, the order of g is ≤ 2.
From 3.1 we know
n/n1 fj zj
gj ξ ρj
n/n1
ρj
ρj ξ ⇒ g ξ,
n/n1
fj zj ρj ξ − afj−n zj ρj ξ − b gj ξ − agj−n ξ − ρj
b ⇒ g ξ − ag −n ξ
3.2
in C \ S locally uniformly with respect to the spherical metric, where S is the set of all poles
of gξ.
If g g n − a ≡ 0, then −1/n 1g n1 ≡ aξ c, where c is a constant. This contradicts with
≡ 0.
g being a meromorphic function. So g g n − a /
0, by Lemma 2.3, then g is also a constant which is a contradiction with g
If g g n − a /
being a nonconstant. Hence, g g n − a is a nonconstant meromorphic function and has at least
one zero.
Journal of Applied Mathematics
5
Next we prove that g g n − a has just a unique zero. On the contrary, let ξ0 and ξ0∗ be two
distinct zeros of g g n − a, and choose δ> 0 small enough such that Dξ0 , δ ∩ Dξ0∗ , δ φ,
where Dξ0 , δ {ξ : |ξ − ξ0 | < δ} and Dξ0∗ , δ {ξ : |ξ − ξ0∗ | < δ}. From 3.2, by Hurwitz s
theorem, there exist points ξj ∈ Dξ0 , δ, ξj∗ ∈ Dξ0∗ , δ such that for sufficiently large j
fj zj ρj ξj − afj−n zj ρj ξj − b 0,
fj zj ρj ξj∗ − afj−n zj ρj ξj∗ − b 0.
3.3
By the hypothesis that, for each pair of functions f and g in F, f − af −n and g − ag −n
share b in D, we know that for any positive integer m
−n
fm
zj ρj ξj − afm
zj ρj ξj − b 0,
−n
zj ρj ξj∗ − afm
zj ρj ξj∗ − b 0.
fm
3.4
−n
Fix m, take j → ∞, and note zj ρj ξj → 0, zj ρj ξj∗ → 0, then fm
0 − afm
0 − b 0.
−n
− afm
− b have no accumulation point, so
Since the zeros of fm
zj ρj ξj 0,
zj ρj ξj∗ 0.
3.5
Hence, ξj −zj /ρj , ξj∗ −zj /ρj . This contradicts with ξj ∈ Dξ0 , δ, ξj∗ ∈ Dξ0∗ , δ, and
Dξ0 , δ ∩ Dξ0∗ , δ φ. So g g n − a has just a unique zero, which can be denoted by ξ0 . By
Lemma 2.4, g is not any transcendental function.
If g is a nonconstant polynomial, then g g n − a Aξ − ξ0 l , where A is a nonzero
constant, l is a positive integer, because l ≥ n ≥ 3. Set φ 1/n1g n1 , then φ Aξ−ξ0 l a
and φ Alξ − ξ0 l−1 . Note that the zeros of φ are of multiplicity ≥ 4. But φ has only one zero
0.
ξ0 , so φ has only the same zero ξ0 too. Hence, φ ξ0 0 which contradicts with φ ξ0 a /
Therefore, g and φ are rational functions which are not polynomials, and φ − a has just a
unique zero ξ0 .
Next we prove that there exists no rational function such as φ. Noting that φ 1/n 1g n1 , we can set
ξ − ξ1 m1 ξ − ξ2 m2 · · · ξ − ξs ms
φξ A n n
n ,
ξ − η1 1 ξ − η2 2 · · · ξ − ηt t
3.6
where A is a nonzero constant, s ≥ 1, t ≥ 1, mi ≥ n 1 ≥ 3 i 1, 2, . . . , s, nj ≥ n 1 ≥ 3 j 1, 2, . . . , t. For stating briefly, denote
m m1 m2 · · · ms ≥ 3s,
N n1 n2 · · · nt ≥ 3t.
3.7
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Journal of Applied Mathematics
From 3.6,
φ ξ Aξ − ξ1 m1 −1 ξ − ξ2 m2 −1 · · · ξ − ξs ms −1 hξ p1 ξ
,
n 1 n 1 n 1
q1 ξ
ξ − η1 1 ξ − η2 2 · · · ξ − ηt t
3.8
where
hξ m − N − tξst−1 ast−2 ξst−2 · · · a0 ,
p1 ξ Aξ − ξ1 m1 −1 ξ − ξ2 m2 −1 · · · ξ − ξs ms −1 hξ,
n 1 n 1 n 1
q1 ξ ξ − η1 1 ξ − η2 2 · · · ξ − ηt t
3.9
are polynomials. Since φ ξ a has only a unique zero ξ0 , set
Bξ − ξ0 l
n 1 n 1 n 1 ,
ξ − η1 1 ξ − η2 2 · · · ξ − ηt t
φ ξ a 3.10
where B is a nonzero constant, so
ξ − ξ0 l−1 p2 ξ
n1 2 n 2 n 2 ,
ξ − η1
ξ − η2 2 · · · ξ − ηt t
φ ξ 3.11
where p2 ξ Bl − N − 2tξt bt−1 ξt−1 · · · b0 is a polynomial. From 3.8 we also have
φ ξ ξ − ξ1 m1 −2 ξ − ξ2 m2 −2 · · · ξ − ξs ms −2 p3 ξ
n 2 n 2 n 2 ,
ξ − η1 1 ξ − η2 2 · · · ξ − ηt t
3.12
where p3 ξ is also a polynomial.
Let degp denote the degree of a polynomial pξ.
From 3.8 and 3.9,
degh ≤ s t − 1,
deg p1 ≤ m t − 1,
deg q1 N t.
3.13
Similarly from 3.11, 3.12 and noting 3.13,
deg p2 ≤ t,
3.14
deg p3 ≤ deg p1 t − 1 − m − 2s ≤ 2t 2s − 2.
3.15
Note that mi ≥ 3 i 1, 2, . . . , s, it follows from 3.8 and 3.10 that φ ξi 0 i 0. Thus, ξ0 /
ξi i 1, 2, . . . , s, and then ξ − ξ0 l−1 is a factor of
1, 2, . . . , s and φ ξ0 a /
Journal of Applied Mathematics
7
p3 ξ. Hence, we get that l − 1 ≤ degp3 . Combining 3.11 and 3.12 we also have m − 2s degp2 l − 1 − degp3 ≤ degp2 . By 3.14 we obtain
m − 2s ≤ deg p2 ≤ t.
3.16
Since m ≥ 3s, we know by 3.16 that
s ≤ t.
3.17
4t − 1 ≤ N t − 1 ≤ l − 1 ≤ deg p3 ≤ 2t 2s − 2.
3.18
If l ≥ N t, by 3.15, then
Noting 3.17, we obtain 1 ≤ 0; a contradiction.
If l < N t, from 3.8 and 3.10, then degp1 degq1 . Noting that degh ≤ s t − 1,
degp1 ≤ m t − 1, and degq1 N t, hence m ≥ N 1 ≥ 3t 1. By 3.16, 2t 1 ≤ 2s. From
3.17, we obtain 1 ≤ 0; a contradiction.
The proof of Theorem 1.7 is complete.
Proof of Theorem 1.11. The proof of this theorem is the same as the proof of Theorem 1.7, some
different places are stated as follows.
The zeros of g are multiple;
l ≥ 2n 1 3.
3.19
The zeros of φ are of multiplicity ≥4:
mi ≥ 2n 1 4 i 1, 2, . . . , s,
nj ≥ n 1 2
m m1 m2 · · · ms ≥ 4s,
j 1, 2, . . . , t ;
N n1 n2 · · · nt ≥ 2t.
3.20
3.7
Noting m ≥ 4s, by 3.16 we have
2s ≤ t.
3.17
3t − 1 ≤ N t − 1 ≤ l − 1 ≤ deg p3 ≤ 2t 2s − 2.
3.21
If l ≥ N t, by 3.15, then
Noting 3.17, we obtain 1 ≤ 0; a contradiction.
If l < N t, from 3.8 and 3.10, then degp1 degq1 . Noting that degh ≤ s t − 1,
degp1 ≤ m t − 1, and degq1 N t, hence m ≥ N 1 ≥ 2t 1. By 3.16, 2t 1 ≤ 2s t.
From 3.17 , we obtain 1 ≤ 0; a contradiction.
The proof of Theorem 1.11 is complete.
8
Journal of Applied Mathematics
Acknowledgment
The authors would like to express their hearty thanks to Professor Qingcai Zhang for
supplying them his helpful reprint. They wish to thank the managing editor and referees
for their very helpful comments and useful suggestions. This work was completed of the
support with the NSF of China 10771220 and Doctorial Point Fund of National Education
Ministry of China 200810780002.
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